e_021230
plain-language theorem explainer
For the six Fin-4 indices (0,2,1,2,3,0), the folded coupling numerator equals eight times the explicit integer kernel entry. Gravity analysts cite it when discharging one cell of the 4D Regge midpoint M2TT identity. The proof is a pure kernel decision: Lean closes the equality by compute.
Claim. Let $a,b,c,d,i,j\in\{0,1,2,3\}$. Writing $N$ for the folded coupling numerator and $Z$ for the explicit integer kernel table, one has $N(0,2,1,2,3,0)=8\,Z(0,2,1,2,3,0)$.
background
The ambient setting is the exact midpoint form of the 4D Regge M2TT identity in the Gravity.Analysis stack. The module is one of the numbered chunks that certify the pointwise relation between two integer-valued kernels on $(\mathrm{Fin},4)^6$.
The numerator $N=\mathrm{m2Num}$ is defined by folding a fixed coupling list: start at $0$ and add, for each coupling term $t$, the contribution of $t$ at the six indices. The comparison object $Z=\mathrm{explicitZ}$ is a closed pattern-match table on those same six indices, returning small integers such as $4$, $-2$, and so on for the nonzero cells.
The local claim is the single cell of the identity $N=8Z$ at indices $(0,2,1,2,3,0)$. Sibling theorems cover the other cells of the same chunk; the module header records that the chunk is discharged by 256 kernel decides.
proof idea
One-line computational proof. Both sides are closed integer terms once the six concrete Fin-4 indices are substituted, so decide evaluates the fold that defines the numerator and the pattern match that defines the explicit kernel, then checks equality in $\mathbb{Z}$. No lemmas beyond the two kernel definitions are required.
why it matters
The parent theorem is the universal assembly statement: for every $a,b,c,d,i,j:\mathrm{Fin},4$, the numerator equals eight times the explicit kernel. That assembly proceeds by exhaustive fin_cases on the six indices and lands on cell theorems such as this one.
In the Recognition gravity stack the identity is bookkeeping for the exact midpoint Regge calculus in four dimensions: it certifies that the folded coupling expression collapses to a sparse explicit table scaled by eight. The result is pure discrete linear algebra on Fin-4 indices; it does not itself invoke the forcing chain (T0–T8), the J-cost, or the phi ladder, but it is part of the certified geometric substrate those layers sit on.
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